       Mathematica Problem

• Subject: Mathematica Problem
• From: jack at sun.acs.udel.edu
• Date: Mon, 19 Jun 89 14:51:49 -0500
• Apparently-to: mathgroup-out at yoda.ncsa.uiuc.edu

```Perhaps someone can help solve the following Mathematica problem:

Given y' = y,

solve using a series solution and prove that the result is E[x].

It is proving that the result is E[x] that I have not been able to do.
Here is a sample session:

In:= Derivative[n_][y][x_]:=Derivative[n-1][y][x]

In:= Series[y[x],{x,0,5}]

2         3         4         5
y x    y x    y x    y x        6
Out= y + y x + ------- + ------- + ------- + ------- + O[x]
2         6        24        120

In:= seriesbyy0[x_,n_]:=Normal[Expand[Cancel[Series[y[x],{x,0,n}]/y]]]

In:= expseries[x_,n_]:=Sum[x^i/i!,{i,0,n}]

In:= seriesbyy0[x,6]

2    3    4    5     6
x    x    x    x     x
Out= 1 + x + -- + -- + -- + --- + ---
2    6    24   120   720

In:= expseries[x,6]

2    3    4    5     6
x    x    x    x     x
Out= 1 + x + -- + -- + -- + --- + ---
2    6    24   120   720

In:= seriesbyy0[x,100]-expseries[x,100]

Out= 0

As shown, I can prove the results are identical for a constant value of
the argument "n", but not for general n.  I realize that the identity is
proven by visual inspection, but I would like to prove it by showing the
identity

seriesbyy0[x,n]  ==  expseries[x,n]

evaluates to true.  I have tried LogicalExpand, but to no avail.

Jack Seltzer
jack at sun.acs.udel.edu

```

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